The problem is the false positive isn't truly independent of environmental factors.

PCR determines how many exponential amplifications are necessary to get a detectable amount of the target RNA sequences.

The amount of that target RNA floating around in the environment does slightly impact the base amount present in even covid-negative patients.

It's true that people who had COVID can still have viral RNA in their noses for a while and still test positive.

The most sensitive tests have a limit of detection of a couple hundred copies/ml, and the collection kits usually use 1ml of VTM or equivalent. I find it hard to believe that a significant number of people will have enough viral RNA in their nose to put hundreds of copies onto a swab unless it's their own body that's producing those copies.

> I find it hard to believe that a significant number of people will have enough viral RNA in their nose to put hundreds of copies onto a swab unless it's their own body that's producing those copies.

This has been confirmed in studies. Researchers have taken samples that are PCR positive at various cycle counts, and tried to culture viruses from them. Here's one:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7427302/

"RT-PCR cycle threshold (Ct) values correlate strongly with cultivable virus. Probability of culturing virus declines to 8% in samples with Ct > 35 and to 6% 10 days after onset; it is similar in asymptomatic and symptomatic persons."

(Note: this is not the only one of these papers that has been published. This is just the first one I found with a quick search.)

> At the latest low point in the epidemic, some areas were seeing test positivity rates of under 0.5%. If false positives were a large portion of positives, this would seem to be impossible.

Its quite possible, especially if there is a testing bias toward people who are less likely to have the disease

Even if they were testing people that were all externally known not to have COVID, if only 0.5% of people are testing positive, that's the false positive rate.

All the data we have indicates that false positives on PCR are extremely rare. The only place it gets fuzzy is the definition of positive. People who were previously infected can test positive, and people with very mild infections can also test positive. People who have never been infected though essentially do not test positive.